1. Field of the Invention
This invention relates to equalizers that estimate the original signal in the presence of noise, delay and interference.
2. Description of the Related Art
If signal x(n) is transmitted through a linear dispersive channel, the received signal y(n) can be modeled by
                              y          ⁡                      (            n            )                          =                                            ∑                              k                =                                  -                  L                                            L                        ⁢                                          a                k                            ⁢                              x                ⁡                                  (                                      n                    -                    k                                    )                                                              +                      e            ⁡                          (              n              )                                                          (        1        )            where e(n) is the additive white Gaussian noise which might be modeled by the following statistics:E[e(n)]=0,E[e(n)e(m)]=σe2δ(n−m).
It is further assumed that signal x(n)is a binary signal (1, −1) and an equi-probable and independent sequence with the following statistics:E[x(n)]=0,E[x(n)x(m)]=σx2δ(n−m).
As can be seen in equation (1), 2L+1 signals {y(n−L), y(n−L+1), . . . , y(n), . . . , y(n+L−1), y(n+L)} contain some information on x(n) and can be used to estimate x(n). These 2L+1 signals can be represented as a vector as follows:Y(n)=[y(n−L),y(n−L+1), . . . ,y(n), . . . ,y(n+L−1),y(n+L)]T.
Furthermore, {x(n−2L), x(n−2L+1), . . . , x(n−1), x(n)} will have some effects on y(n−L) and {x(n), x(n+1), . . . , x(n+2L−1), x(n+2L)} on y(n+L). Thus, it can be said that {x(n−2L), x(n−2L+1), . . . , x(n), . . . , x(n+2L−1), x(n+2L)} affect the estimation of x(n) at the receiver. As stated previously, input vector X(n) and noise vector N(n) are defined as follows:X(n)=[x(n−2L),x(n−2L+1), . . . ,x(n), . . . ,x(n+2L−1),x(n+2L)]T N(n)=[e(n−L),e(n−L+1), . . . ,e(n), . . . ,e(n+L−1),e(n+L)]T.
It is noted that the dimension of the input vector, X(n), is 4L+1. This analysis can be easily extended to non-symmetric channels.
Equalization has been important in communications and data storage, and numerous algorithms have been proposed. Among various equalization methods, linear equalization has been widely used due to its speed and simplicity. The linear equalizer is frequently implemented using the LMS algorithm as follows:z[n]=WT(n)Y(n)where Y(n)=[y(n−L),y(n−L+1), . . . , y(n), . . . , y(n+L−1),y(n+L)]T, z[n] is an output of the equalizer, and W(n)=[w−L, w−L+1, . . . , w, . . . , wL−1, wL]T is a weight vector. The weight vector is updated as follows:W(n+1)=W(n)+cλY(n)where λ is the learning rate, c is 1 if signal 1 is transmitted and -1 if signal -1 is transmitted.
In the present invention, the linear equalizer is implemented utilizing second-order statistics considering equalization as a classification problem. As a result, the convergence speed and the performance are significantly improved.